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On bounded pseudodifferential operators in Wiener spaces
Laurent Amour, Lisette Jager, Jean Nourrigat
To cite this version:
Laurent Amour, Lisette Jager, Jean Nourrigat. On bounded pseudodifferential operators in Wiener spaces. Journal of Functional Analysis, Elsevier, 2015, 269 (9), pp.2747 - 2812.
�10.1016/j.jfa.2015.08.004�. �hal-01881667�
On bounded pseudodifferential operators in Wiener spaces
Laurent Amour, Lisette Jager and Jean Nourrigat Universit´ e de Reims
Dedicated to the memory of Bernard Lascar Abstract
We aim at extending the definition of the Weyl calculus to an infinite dimensional setting, by replacing the phase spaceR2nbyB2, where (i, H, B) is an abstract Wiener space. A first approach is to generalize the integral definition using the Wigner function. The symbol is then a function defined on B2 and belonging to aL1space for a gaussian measure, the Weyl operator is defined as a quadratic form on a dense subspace of L2(B). For example, the symbol can be the stochastic extension on B2, in the sense of L. Gross, of a functionF which is continuous and bounded onH2.
In the second approach, this function F defined onH2 satisfies differentiability conditions analogous to the finite dimensional ones. One needs to introduce hybrid operators acting as Weyl operators on the variables of finite dimensional subset ofH and as Anti-Wick operators on the rest of the variables. The final Weyl operator is then defined as a limit and it is continuous on aL2 space. Under rather weak conditions, it is an extension of the operator defined by the first approach.
We give examples of monomial symbols linking this construction to the classical pseudodifferential oper- ators theory and other examples related to other fields or previous works on this subject.
Contents
1 Introduction 2
2 Anti-Wick, Weyl and hybrid operators. 7
2.1 Heat semigroup and anti-Wick operator . . . 7
2.2 Wiener measure decomposition . . . 8
2.3 Partial heat semigroup and hybrid operators . . . 8
3 Plan of the proof of Theorem 1.4 10 4 Some useful operators in Wiener spaces 12 4.1 Abstract Wiener spaces . . . 12
4.2 Stochastic extensions . . . 13
4.3 The Heat operator (continued) . . . 14
4.4 Coherent states and Segal Bargmann transformation . . . 16
4.5 Wigner Gaussian function . . . 18
4.6 Convergence in Definition 1.2. . . 20
4.7 Anti-Wick and hybrid operators . . . 21
4.8 Partial Heat semigroups and stochastic extensions. . . 23
5 Proof of Proposition 3.1. 24
6 Proof of Proposition 3.3. 28
7 Wick symbol of a Weyl operator 30
8 Examples 31 8.1 Examples of Wiener spaces . . . 31 8.2 Examples of stochastic extensions. . . 33 8.3 Examples of symbols inSm(M, ε). . . 38
9 A covariance result 42
10 Aknowledgments 45
1 Introduction
The Weyl calculus associates with every functionF, called symbol, bounded and measurable onR2n and every h >0 an operator, denoted byOpW eyl,Lebh (F), from the Schwartz spaceS(Rn) into its dual spaceS0(Rn). For a functionf belonging toS(Rn), this operator is, formally, defined by
(OpW eyl,Lebh (F)f)(x) = (2πh)−n Z
R2n
ehi(x−y)·ξF
x+y 2 , ξ
f(y)dλ(y, ξ) (1)
whereλis the Lebesgue measure. IfF isC∞ and bounded onR2nas well as all its derivatives, then Calder´on- Vaillancourt’s theorem states thatOpW eyl,Lebh (F) extends to a bounded operator onL2(Rn, λ) (see [C-V], [HO]).
The definition of the Weyl operator as an application onS(Rn) with values inS0(Rn) or, equivalently, as a quadratic form onS(Rn), has already been extended to the infinite-dimensional case for some specific symbols by Kree-R¸aczka [K-R] and, up to a small modification, by Bernard Lascar ([LA-1]), (see [LA-2] to [LA-10] as well for applications). In the present paper the hypotheses on the symbol of the operator (the function F in (1)) are weaker than by these authors. We also give a Calder´on-Vaillancourt type result in this context.
The classical definition (1) does not lend itself to an extension to an infinite dimensional case. We shall use instead the definition of OpW eyl,Lebh (F) in which the Wigner function appears. This operator is the only one that satisfies, for allf andg inS(Rn):
< OpW eyl,Lebh (F)f, g >L2(Rn,λ)= (2πh)−n Z
R2n
F(Z)HhLeb(f, g, Z)dλ(Z), (2) whereHhLeb(f, g, .) is the Wigner function (for the Lebesgue measure):
HhLeb(f, g, Z) = Z
Rn
e−hit·ζf
z+ t 2
g
z−t
2
dλ(t) Z = (z, ζ)∈R2n (3) (cf Unterberger [U-2], or Lerner [LE], sections 2.1.1 and 2.1.2, or Combescure Robert [C-R], section 2.2).
The expression (2) is well defined for f andg in S(Rn) even if F is only supposed to be measurable and bounded. Indeed, the Wigner function HhLeb(f, g,·) is in S(R2n). We first extend this definition (2) to the infinite dimensional case (using Wiener spaces) in the same spirit, by defining the operator as a bilinear form on a convenient dense subspace, with very weak assumptions on F (weaker than those of [LA-1] and [K-R]) : see definition 1.2 below. To this aim we associate a Gaussian Wigner function HhGauss(f, g) with every couple (f, g), wheref andgare convenient functions defined on a Wiener space. Then we prove that, under Calder´on- Vaillancourt type conditions, this operator extends to a bounded operator on aL2 space (Theorem 1.4, which is the main result).
We successively indicate what replaces the space Rn, the Lebesgue measure, the Schwartz’s space S(Rn) and the Wigner function. Then we shall be ready to define the Weyl operator as a quadratic form, as in (2).
The spaceRn is replaced by a real separable infinite-dimensional Hilbert spaceH, the configuration space.
The symbolF is a function on the phase spaceH2. One denotes bya·bthe scalar product of two elementsa andbofH and by|a|the norm of an elementaofH.
The Lebesgue measure is replaced by a Gaussian measure associated with the norm of H. But, since H is infinite-dimensional, this Gaussian measure will be a measure, not on H (which is impossible) but on a convenient Banach spaceB containingH. Abstract Wiener spaces are an adequate frame. An abstract Wiener space is a triple (i, H, B), where H is a real, separable Hilbert space, B a Banach space and i a continuous injection from H into B such that i(H) is dense in B, other conditions being satisfied (see [G-2, G-3, K] or Definition 4.1). The injectioniis generally not mentioned and one usually identifies H with its dual space, so that the preceding hypotheses imply
B0⊂H0 =H ⊂B, (4)
where every space is continuously embedded as a dense subspace of the following one. For everyuinB0 andx inB, one denotes byu(x) the duality between these elements and one supposes that, ifxis inH,u(x) =u·x, This will be the case in the rest of the article.
One denotes by F(H) (resp. F(B0)) the set of all finite-dimensional subspaces E ofH (resp. of B0). We note thatF(B0)⊂ F(H) by (4). For everyEbelonging toF(H) and every positiveh, one defines a probability measureµE,h onEby:
dµE,h(x) = (2πh)−(1/2)dim(E)e−|x|
2
2h dλ(x), (5)
where λis the Lebesgue measure on E, the norm onE being the restriction to E of the norm of H. If B is a Banach space satisfying (4), the continuity and density conditions being satisfied, one defines, for eachE is F(B0), an applicationPE:B →E by:
PE(x) =
n
X
j=1
uj(x)uj, (6)
where {u1, ..., un} is a basis of E, orthonormal for the restriction to E of the scalar product of H. This application does not depend on the choice of the basis. If the Banach space B satisfies proper conditions, (see Definition 4.1 for precisions), for each positiveh, one can derive a probability measure µB,h on the Borel σ−algebra ofB, with the following property. For everyE in F(B0) and every functionϕ in L1(E, µE,h), the functionϕ◦PE is in L1(B, µB,h), and one has
Z
B
(ϕ◦PE)(x)dµB,h(x) = Z
E
ϕ(y)dµE,h(y). (7)
Other properties of the measure µB,h are recalled in section 4.1 and examples are given in Section 8. In Section 4.2, the same notions will be seen for subspacesE inF(H). The Banach space Bassociated withH is not unique but for any choice ofB and any positiveh, the spaceL2(B, µB,h) is isomorphic to the symmetrized Fock space Fs(H), which does not depend on the choice of B (cf [F] or [A-J-N-1]). If the Hilbert space H is finite dimensional, then B=H. If not,B is sometimes derived fromH thanks to a Hilbert-Schmidt operator (cf [G-3], Example 2, p. 192)), but other constructions are possible. In Section, 8 examples related to Brownian motion, field theory and interacting lattices will be given.
Now let us introduce the space which will replaceS(Rn) in (2) (3). For everyEin F(H), we shall need the isometric isomorphismγE,h/2 betweenL2(E, µE,h/2) andL2(E, λ) given by
(γE,h/2ϕ)(x) = (πh)−(1/4)dim(E)ϕ(x)e−|x|
2
2h ϕ∈L2(E, µE,h/2). (8)
One denotes by SE the space of all functionsϕ:E →Csuch thatγE,h/2ϕbelongs to the Schwartz space of rapidly decreasing functionsS(E). ForE∈ F(B0), letDE be the set of applicationsf :B→C of the form f =ϕ◦PE, where PE :B → E is defined by (6) and ϕ: E →C belongs toSE. We denote byD the union
of the spaces DE, taken over all E in F(B0). This space D is dense inL2(B, µB,h/2). Indeed, if (ej)(j≥1) is a Hilbert basis ofH, the vectors ej belonging to B0, the set of functions onB which are polynomials of a finite number of functionsx→ej(x) is contained inDand is dense inL2(B, µB,h/2) (see for example [A-J-N-1]).The constant functions belong toDE for every E inF(B0).
We very often need the following classical result (Kuo, [K], Chapter 1 Section 4).
Proposition 1.1.
1. The spaceB0 is contained inL2(B, µB,h)and the norm ofu∈B0, considered as an element ofL2(B, µB,h), is equal to √
h|u|.
2. The inclusion ofB0 intoL2(B, µB,h)extends to a continuous linear map from H intoL2(B, µB,h), with norm√
h, denoted byu→`u.
The first point can be seen by applying (7) to a one dimensional spaceE. The map`turnsHinto a Gaussian space in the sense of [J] or into a ”Gaussian random process”, in the sense of [SI]. In the case of Example 8.2, for everyuin the Cameron-Martin space,`uis the Itˆo integral of the function u0.
Let us now define the Wigner-Gauss function, which will replace the usual Wigner function. Let (i, H, B) be a Wiener space satisfying (4). For every subspaceE in F(H), for allϕandψin SE, one defines a function HbhGauss(ϕ, ψ) onE2, setting:
HbhGauss(ϕ, ψ)(z, ζ) =e|ζ|
2 h
Z
E
e−2hiζ·tϕ(z+t)ψ(z−t)dµE,h/2(t) (z, ζ)∈E2. (9) One notices that, forZ in E2:
HbhGauss(ϕ, ψ)(Z) = 2−dim(E)e|Z|
2
h HhLeb(γE,h/2ϕ, γE,h/2ψ)(Z) Z ∈E2. (10) One will see (Proposition 4.8) that this function belongs toL1(E2, µE2,h/2) and toL2(E2, µE2,h/4) as well. For all functionsf andg in DE, (E in F(B0)), of the formf =fb◦PE and g =bg◦PE, withfbandbg in SE, one defines the Wigner-Gauss transformHhGauss(f, g), which is the function defined onE2 by:
HhGauss(f, g)(Z) =HbhGauss(f ,bbg)(PEZ) a.eZ∈B2. (11) One writesPEinstead ofPE2 for the sake of simplicity. One will see in Section 4.2 how to modify this definition if E ∈ F(H). According to (7), it follows from Proposition 4.8 that this function is in L1(B2, µB2,h/2) and L2(B2, µB2,h/4). Iff andgare inD, the subspaceEsuch thatf andgare inDEis not unique, but the function defined above does not depend onE. Proposition 4.8 states that this Wigner-Gauss transformation extends, by density, fromL2(B, µB,h/2)×L2(B, µB,h/2) toL2(B2, µB2,h/4) and to the space of continuous functions defined onH2. One will see, in (76), another expression ofHhGauss(f, g), using Segal-Bargmann transforms off andg.
Now we are almost ready to define the Weyl operator associated with a symbol F. If (i, H, B) is a Wiener space, we have the choice of two phase spaces: H2 anB2. The first one is equipped with the symplectic form σ((x, ξ),(y, η)) =y·ξ−x·η, but not with a measure adapted to the integrations we want to conduct. On the contrary, the spaceB2is equipped with the Gaussian measureµB2,h, but it is not a symplectic space.
This difficulty is overcome in the following way. The symbols will be initially defined as functions on H2. There exists an operation, introduced by L. Gross ([G-1], see Ramer [RA]) and usually called stochastic extension, which associates with a Borel function F on H or H2 a Borel function Fe on B or on B2. This stochastic extension, which is not a genuine extension, will be recalled in Definition 4.4. What will appear in the initial definition formula (13) of the Weyl operators is the stochastic extensionFe of the initial symbolF.
In the second step, concerned with the bounded extension in L2(B, µB,h/2) of this initial operator, we shall restrict ourselves to bounded symbols. But in the initial definition of the operator as a quadratic form on D, a polynomial growth will be enough. For the initial definition (13), the functionFe (stochastic extension of the symbolF) will be inL1(B2, µB2,h/2) and the polynomial growth will be expressed in terms of the existence of a nonnegative integermsuch that the following norm is defined.
Nm(Fe) = sup
Y∈H2
kFe(·+Y)kL1(B2,µB2,h/2)
(1 +|Y|)m . (12)
This norm is finite if the functionF is bounded or if it is a polynomial expression of degreemwith respect to functions (x, ξ)→`a(x) +`b(ξ), withaandbin H.
Definition 1.2. Let (i, H, B) be an abstract Wiener space satisfying (4) and h > 0. Let Fe be a function in L1(B2, µB2,h/2). Suppose there existsm≥0 such that the norm Nm(Fe)is finite. We define a quadratic form QW eyl(Fe)onD × D in the following way. For allf andg in D, one sets:
QW eylh (F)(f, g) =e Z
B2
Fe(Z)HhGauss(f, g)(Z)dµB2,h/2(Z) (13) whereHhGauss(f, g)is defined in (11).
The convergence will be proved in Proposition 4.10. If Fe is bounded, the convergence is a consequence of Proposition 4.8, since HhGauss(f, g) is in L1(B2, µB2,h/2). If we only suppose that Nm(Fe) is finite, other arguments are necessary.
One sees the relationship with the classical definition (2), (3). IfF has the formF =Fb◦PE, whereE is in F(B0) andFbis a measurable bounded function on E2, iff =fb◦PE andg=bg◦PE, wherefbandbgare in SE, one has
QW eylh (Fb◦PE)(fb◦PE,bg◦PE) =< OpW eyl,Lebh (F)γb E,h/2f , γb E,h/2bg >L2(E,λ) (14) whereOpW eyl,Lebh (F) is defined onb E as it is in (2), (3) onRn. This equality comes from (10).
One can compare this definition with the definition in [K-R] or [LA-1]. The authors define an anti-Wick operator associated with a symbol, a function defined onH2which is, for example, the Fourier transform of a complex measure bounded on H2. In [K-R] they associate, too, with such a symbol, a Weyl operator defined as a quadratic form on a dense subspace. When the symbol F is the Fourier transform of a complex measure bounded on H2, we prove, in Proposition 8.3, that F admits a stochastic extensionF, with which Definitione 1.2 above associates a Weyl operator, defined as a quadratic form. For this kind of symbols, the Weyl operator (as a quadratic form) is explicitly written in (139) and (135) or, equivalently, in (139) and (141). This last form can be found in [K-R] as well. The anti-Wick operator can be found in [K-R, LA-1] in the form (140)-(135).
Our Definition 1.2 is more general, insofar as admitting a stochastic extension is more general than being the Fourier transform of a bounded measure. Section 8.2 gives other examples of classes of functions admitting stochastic extensions.
However, all functions do not have a stochastic extension. If H is infinite dimensional, the norm function, which associates with everyx∈H its norm|x|, admits no stochastic extension (see Kuo, [K], Chapter 1, section 4). In the same way, the functionx→e−|x|2, defined onH, has no stochastic extension. We shall not be able to define a pseudodifferential operator, whose symbol would be F(x, ξ) = e−|x|2. This operator would have to be the multiplication by e−|x|2, but this function, lacking an extension, is only defined on a negligible set (the Hilbert space H is negligible in B, cf [K]) and this does not make sense. On the contrary, the function t → e−(At)·t, where the operatorA is positive, symmetrical and trace-class on H, has a stochastic extension (see Proposition 8.7).
It now remains to extend the bilinear form defined above on the subspaceDto get a linear operator bounded on L2(B, µB,h/2). When the symbol is the Fourier transform of a measure bounded on H2 (case treated in [K-R]), the upper bound on the norm is a consequence of Proposition 8.10. For other cases we must specify the hypotheses on F, which will strongly depend on the choice of a Hilbert basis of H. We can now state the hypothesis on the functionF, which will be the symbol in our version of the Calder´on-Vaillancourt theorem.
Definition 1.3. Let (i, H, B) be an abstract Wiener space satisfying (4). The norm of H will hereafter be denoted by | · | and the scalar product of two elements aand b of H will be denoted by a·b. The norm of an element ofH2is denoted by|·|as well. For allX= (x, ξ)andY = (y, η)inH2, we setσ(X, Y) =y·ξ−x·η. We choose a Hilbert basis(ej)(j∈Γ) ofH, each vector belonging toB0, indexed by a countable setΓ. Setuj= (ej,0) andvj= (0, ej) (j ∈Γ). A multi-index is a map (α, β)from Γ intoN×N such that αj =βj = 0except for a finite number of indices. LetM be a nonnegative real number,ma nonnegative integer andε= (εj)(j∈Γ)a family of nonnegative real numbers. One denotes by Sm(M, ε) the set of bounded continuous functionsF : H2 →C satisfying the following condition. For every multi-index (α, β) such that0 ≤αj ≤m and 0≤βj ≤m for all j∈Γ, the following derivative
∂uα∂βvF=
Y
j∈Γ
∂uαjj∂vβjj
F (15)
is well defined, continuous onH2 and satisfies, for every(x, ξ)in H2
Y
j∈Γ
∂αuj
j∂vβj
j
F(x, ξ)
≤M Y
j∈Γ
εαjj+βj . (16)
The choice of indexing the basis (ej) by an arbitrary countable set Γ is motivated by possible applications in lattice theory. Cordes [C], Coifman Meyer [C-M] (for the standard quantization, not for the Weyl one) and Hwang [HW] remarked that, in the Calder´on-Vaillancourt bounds, it is enough to consider multi-indices (α, β) such that 0≤αj ≤1 and 0≤βj ≤1 for everyj, which inspired the definition ofSm(M, ε). One finds in Section 8 examples of functions ofSm(M, ε) coming from interacting lattices models.
The main result can be stated as follows.
Theorem 1.4. Let(i, H, B)be an abstract Wiener space satisfying (4) andhbe a positive number. Let(ej)(j∈Γ) be a Hilbert space basis ofH, each vector belonging toB0, indexed by a countable setΓ. LetF be a function on H2 satisfying the following two hypotheses.
(H1) The function F belongs to the classS2(M, ε) of Definition 1.3, where M is a nonnegative real number andε= (εj)(j∈Γ) a square summable family of nonnegative real numbers.
(H2) We assume that F has a stochastic extension Fe with respect to both measures µB2,h and µB2,h/2 (see Definition 4.4).
Then there exists an operator, denoted by OpW eylh (F), bounded inL2(B, µB,h/2), such that, for allf andg in D
< OpW eylh (F)f, g >=QW eylh (Fe)(f, g), (17) where the right hand side is defined by Definition 1.2. Moreover, ifhis in (0,1]:
kOpW eylh (F)kL(L2(B,µB,h/2))≤M Y
j∈Γ
(1 + 81πhSεε2j), (18)
where
Sε= sup
j∈Γ
max(1, ε2j). (19)
One will see in Proposition 8.4 that, if a function F is in S1(M, ε) and if the sequence ε = (εj)(j∈Γ) is summable, then F satisfies the hypothesis (H2). Proposition 8.11 gives an example of function in Sm(M, ε), inspired by the lattice theory.
The operatorOpW eylh (F) associated withF ∈S2(M, ε) will not be defined by an integral expression, but as the limit, inL(L2(B, µB,h/2)), of a sequence of operators. In order to define this sequence we shall associate with each subspace E in F(B0), an operator denoted by Ophyb,Eh (Fe) and bounded on L2(B, µB,h/2). This operator behaves as a Weyl operator on a set of variables and as an anti-Wick operator on the other variables.
It will have the formQW eylh (G), whereGis obtained from the extensionFeby applying a partial heat operator concerning the variables in E⊥ (see Section 2). Then we shall prove that, if one replaces E by a sequence E(Λn) =V ect(ej)(j∈Λn)where (Λn) is an increasing sequence of finite subsets of Γ whose union is Γ, then the sequence of operators (Ophyb,E(Λh n)(G))n is a Cauchy sequence inL(L2(B, µB,h/2)). Its limit will be denoted by OpW eylh (F). We shall see that, if one restricts this operator to get a bilinear form on D, it coincides with the one defined by Definition 1.2. In particular it does not depend on the Hilbert basis (ej) chosen to construct it, nor on the sequence (Λn).
The hybrid operator associated with each finite dimensional subspace is defined in Section 2. Section 3 presents more precisely the main steps of the proof of Theorem 1.4. Section 4 first recalls classical facts about Wiener spaces, then gives a precise definition of the stochastic extensions, recalls the Segal Bargmann transfor- mation, gives the necessary upper bounds about the Gaussian Wigner function and gives another expression of the Weyl and hybrid operators, more convenient to get norm estimates. Section 5 establishes the norm estimates which prove that the sequence constructed to approachOpW eylh (F) is indeed a Cauchy sequence. In Section 6 we prove thatOpW eylh (F) is really an extension of the initially defined operator (Definition 1.2). Section 7 studies the Wick symbol of this operator. Section 8 gives examples of Wiener spaces, stochastic extensions and symbols belonging toSm(M, ε).
Some classical properties of the Weyl calculus can be extended in the frame described above. For example the covariance with respect to translation (section 9) or a Beals type characterization ([B1, B2]) for the operators of Theorem 1.4, by norm estimates of their commutators with position and impulsion operators (see [ALN]). As a corollary of this characterization, one can prove that the product of two such operators is still in the same class but this result does not give any further information concerning the symbol of the operator. An asymptotic expansion of the symbol of the product is given in [AN] in the case of the high, but finite dimension.
2 Anti-Wick, Weyl and hybrid operators.
2.1 Heat semigroup and anti-Wick operator
In an Euclidean finite dimensional spaceE one may associate with any Borel bounded functionF onE2, and for allh >0, an operator OpAW,Lebh (F) called the anti-Wick or Berezin-Wick operator. Instead of referring to the usual definition, let us say that it is the only operator satisfying
< OpAW,Lebh (F)f, g >Leb=< OpW eyl,Lebh (eh4∆F)f, g >Leb (20) for allf andgin the Schwartz spaceS(E) ( ∆ =P ∂2
∂x2i +∂ξ∂22 i
being a negative operator). This operator has a bounded extension inL2(E, λ), also denoted by OpAW,Lebh (F), which verifies
kOpAW,Lebh (F)kL(L2(E,λ))≤ kFkL∞(E2). (21) This property is easier to extend and hence is more convenient as a starting point than the usual definition.
See [F] or [LE], Chapter 2.
To extend Definition (20) to the infinite dimensional setting, we need an analogue of the heat semigroup on B2, that is the family of operatorsHet defined by:
(HetF)(X) = Z
B2
F(X+Y)dµB2,t(Y) X ∈B2. (22) The operatorHetcorresponds, in the finite dimensional case, to the operatore2t∆. For everyt >0, the operator Hetis bounded in the space of all bounded Borel functions onB2 (see Hall [HA], Cecil-Hall [C-H]). Moreover, it is bounded and with a norm smaller than 1 from Lp(B2, µB2,t+h) in Lp(B2, µB2,h) (1≤p <∞, t >0,h >0), see Proposition 4.5 below.
For every positiveh, one associates with each bounded Borel functionF onB2an operator called anti-Wick operator. It will first be defined as a quadratic form onD × D, denoted byQAWh (F) and such that
QAWh (F)(f, g) =QW eylh (Heh/2F)(f, g), (23) where the Weyl quadratic form is the one of Definition 1.2. One will see in Corollary 4.12 an expression which is closer to the usual definition, as well as the fact that this quadratic form is linked with a bounded operator, whose norm will be specified.
The following step is associating, with every subspace E in F(B0), a hybrid operator, with the help of a partial heat operator. This hybrid operator behaves as a Weyl operator associated with F regarding some functions, and as an anti-Wick operator associated with the same symbol regarding some other functions.
2.2 Wiener measure decomposition
The following proposition proved in Gross [G-4] or in [RA] allows to split the variables for our hybrid operators.
Proposition 2.1. Let (i, H, B) be an abstract Wiener space. In the following, the injection i is implicitly understood, H is identified with its dual space in such a way that (4) is verified and we fix a subspace E in F(B0). LetE⊥ be the following space:
E⊥={x∈B, u(x) = 0 ∀u∈E}. (24)
Let i1 denote the injection of E⊥∩H in E⊥. Then:
1. The system(i1, E⊥∩H, E⊥)is an abstract Wiener space. We denote by µE⊥,hthe Gaussian measure of parameter honE⊥.
2. We have, for allh >0:
µB,h=µE,h⊗µE⊥,h. (25)
The map x→(PE(x), x−PE(x)) (where PE is defined in (6)) is a bijection between B andE×E⊥. We denote byx= (xE, xE⊥) the variable in B and by X = (XE, XE⊥) the variable in B2. Sometimes, we shall writeX =XE+XE⊥.
2.3 Partial heat semigroup and hybrid operators
IfEis inF(B0), one can define, as in (22), on the one hand a heat semigroup acting only on the variables inE and, on the other hand, another one acting on the variables ofE⊥ (defined in Proposition 2.1). There are two kinds of operators acting on the variables in E: the first kind acts on a space of functions defined onH, the second kind, on a space of functions defined onB. For the operator acting on the variables ofE⊥, the second version only is available.
LetE be inF(B0). For allt >0, we define an operatorHE,t on the space of bounded continuous functions onH2, setting, whenF is such a function:
(HE,tF)(X) = Z
E2
F(X+YE)dµE2,t(YE) X ∈H2. (26) We define likewise an operatorHeE,t, defined by the same formula, but acting on the space of bounded Borel functions onB2.
We may also define a partial heat semi-group, only acting on the variables lying in the subspaceE⊥. For all bounded Borel functions F onB2 and for all positivet, one may define a functionHeE⊥,tF onB2 by setting, for allX in B2:
(HeE⊥,tF)(X) = Z
(E⊥)2
F(X+YE⊥)dµ(E⊥)2,t(YE⊥). (27)
The operators HE,t andHeE,t will mainly appear in Section 3. TheHeE⊥,t will be used now and in Section 6.
We are now ready to define the hybrid operator.
Definition 2.2. If F is a bounded Borel function on B2 and if E is in F(B0), we denote by Qhyb,Eh (F) the quadratic form on D × Dsuch that, for allf andg inD, we have:
Qhyb,Eh (F)(f, g) =QW eylh (HeE⊥,h/2F)(f, g). (28) The following result underlines the relationship between hybrid operators when associated with the same symbol, but with two different subspaces, one being included into the other one.
Let F be a Borel function bounded onB2. Let E1 andE2 be in F(B0), such thatE1 ⊂E2. LetS be the orthogonal complement ofE1 inE2 (for the scalar product ofH). Then, we have:
Qhyb,Eh 1(F) =Qhyb,Eh 2(HeS,h/2F) (29) and we have:
HeS,h/2F(X) = (πh)−dim(S) Z
S2
e−|Y|
2
h F(X+Y)dλ(Y) X ∈B2. (30)
If{e1, ..., en}is an orthonormal basis of the orthogonal complementSofE1inE2, denoting byDjthe subspace ofH spanned byej, we have:
HeS,h/2F =
Y
j≤n
HeDj,h/2
F (31)
whereHeDj,h/2is defined in (26).
The proof of (29) and (30) relies on the Definition 2.2:
Qhyb,Eh j(F) =QW eylh (HeE⊥
j,h/2F) j= 1,2
and on the following equality:
HeE⊥
1,h/2=HeE⊥
2,h/2HeS,h/2.
3 Plan of the proof of Theorem 1.4
Let (ej)(j∈Γ)be a Hilbert basis ofH, as in Definition 1.4. For everyj in Γ, we denote byDj the subspace ofH spanned by ej. For every positive t, let HDj,t andHeDj,t be the operators defined in (26), the first one acting on functions on H2, the second one, on functions on B2. The integration domain in (26) isD2j, which is the subspace ofH2 spanned by (ej,0) and (0, ej). For every finite subsetI of Γ, set:
TeI,h=Y
j∈I
(I−HeDj,h/2), SeI,h=Y
j∈I
HeDj,h/2. (32)
These operators act in the space of bounded Borel functions onB2. We denote byTI,h the operator defined as in (32), but acting in the space of bounded continuous functions onH2.
For every finite subset Iof Γ, let E(I) be the subspace ofH spanned by theej (j∈I).
E(I) = Vect{(ej), j ∈Γ}. (33)
The main Theorem 1.4 is a consequence of the Propositions 3.1 et 3.3 below, which will be proved respectively in Sections 5 and 6.
Proposition 3.1. Let F be in S2(M, ε). We assume that F has a stochastic extension Fe for the measure µB2,h. For every finite subsetI inΓ, for all hin(0,1], there exists a bounded operator, which will be denoted byOphyb,E(I)h (TeI,hF)e such that, for all f andg inD, with the notations (28), (32) and (33):
Qhyb,E(I)h (TeI,hFe)(f, g) =< Ophyb,E(I)h (TeI,hFe)f, g > . (34) Moreover its norm satisfies:
kOphyb,E(I)h (TeI,hFe)kL(L2(B,µB,h/2)) ≤M(81πhSε)|I|Y
j∈I
ε2j. (35)
If one admits Proposition 3.1 (which is a consequence of the combined Propositions 5.2 and 5.3), one can prove the following result.
Proposition 3.2. Let F be a function in S2(M, ε), where the family(εj)(j∈Γ) is square summable. Seth >0.
We assume that F has a stochastic extension Fe for the measure µB2,h. Then, for every increasing sequence (Λn)of finite subsets inΓ, whose union isΓ, there exists a sequence of operators, denoted by(Ophyb,E(Λh n)(Fe)), such that, with the notations (28) and (33), for all f, gin D,:
Qhyb,E(Λh n)(Fe)(f, g) =<(Ophyb,E(Λh n)(F))f, g >e (36) Moreover, the sequence of operators(Ophyb,E(Λh n)(Fe))(n≥1)is a Cauchy sequence inL(L2(B, µB,h/2)). Its limit, denoted byOpW eylh (F), satisfies (18).
Proof. We have, for every continuous and bounded function GonB2, for any finite subset Λ in Γ:
G=X
I⊆Λ
TeI,hSeΛ\I,hG (37)
The sum runs over all the subsets in Λ, including the empty set and Λ itself. As a consequence, the equality (36) will be satisfied if we set:
Qhyb,E(Λ)h (Fe) =X
I⊆Λ
Qhyb,E(Λ)h (TeI,hSeΛ\I,hFe). (38)
From (29) and (30), applied with subspaces E(I) andE(Λ):
Qhyb,E(Λ)h (F) =e X
I⊆Λ
Qhyb,E(Ih )(TeI,hFe). (39) The term corresponding to I = ∅ is the anti-Wick quadratic form associated with Fe and adapted to the Gaussian measure. Therefore, according to Proposition 3.1, there exists a bounded operator, denoted by Ophyb,E(Λ)h such that (36) is satisfied. If (Λn) is an increasing sequence of finite subsets of Γ, we then have, if m < n:
Ophyb,E(Λh n)(F)e −Ophyb,E(Λh m)(Fe) = X
I∈P(m,n)
Ophyb,E(I)h (TeI,hFe) (40) where P(m, n) is the family of subsetsI in Γ, included in Λn, but with at least one element not belonging to Λm. From (40) and from Proposition 3.1, we have, whenm < n:
kOphyb,E(Λh n)(F)e −Ophyb,E(Λh m)(Fe)kL(L2(B,µB,h/2)) ≤ X
I∈P(m,n)
kOphyb,E(I)h (TeI,hF)ke L(L2(B,µB,h/2))
≤M X
I∈P(m,n)
(81πhSε)|I|Y
j∈I
ε2j.
As a consequence, ifm < n:
kOphyb,E(Λh n)(F)e −Ophyb,E(Λh m)(Fe)kL(L2(B,µB,h/2))≤M81πhSε
X
j∈Λn\Λm
ε2j
Y
k∈Λn
(1 + 81πhSεε2k).
If the family (ε2j)(j∈Γ) is summable, the above right hand-side product stays bounded independently of n, whereas the sum tends to 0 when m → +∞. As a consequence, the sequence (Ophyb,E(Λh n)(Fe)) is a Cauchy sequence inL(L2(B, µB,h/2)). Likewise:
kOphyb,E(Λh n)(Fe)kL(L2(B,µB,h/2))≤M Y
k∈Λn
(1 + 81πhSεε2k).
Therefore, the limit of this sequence of operators, denoted byOpW eylh (F), verifies (18).
One could think that the operatorOpW eylh (F) depends on the sequence (Λn), but the following proposition proves that it is not the case. It will be proved in Section 6.
Proposition 3.3. Let F belong to S2(M, ε), where the sequence (εj)(j∈Γ) is square summable. Set h > 0.
Define the function Fe on B2 as the stochastic extension ofF both for the measure µB2,h and for the measure µB2,h/2. Let(Λn)be an increasing sequence of finite subsets of Γ, whose union is Γ. Then, we have, for every f andg in D, settingEn=E(Λn)
n→+∞lim QW eylh (HeE⊥
n,h/2Fe)(f, g) =QW eylh (Fe)(f, g) (41) One can notice that, for a given functionF in S2(M, ε), the stochastic extension (for the measureµB2,s)Fe is unique, but only up to a µB2,s- negligible set. Nevertheless, ifFe andGe are two stochastic extensions of the same functionF, for the measuresµB2,h and µB2,h/2, one can check that HeE⊥
n,h/2Fe and HeE⊥
n,h/2Ge are equal almost everywhere for the measureµB2,h/2.
End of the proof of Theorem 1.4. Once Proposition 3.2 has been established, it only remains to prove (17).
Now for allf andgin Done has, settingEn=E(Λn):
< OpW eylh (F)f, g >= lim
n→+∞<(Ophyb,Eh n(Fe))f, g >= lim
n→+∞QW eylh (HeE⊥
n,h/2Fe)(f, g).
Hence the equality (17) of Theorem 1.4 comes from the above equality and from (41).
4 Some useful operators in Wiener spaces
We first recall the precise definition of Wiener spaces as well as some of their properties. Then we adapt some classical notions for Wiener spaces : coherent states, Segal-Bargmann transformation. Next we give properties of the Gaussian Wigner function of Section 1. This will allow us to write Definition 1.2 of the Weyl operators and Definition 2.2 of the hybrid operators in a way more suitable for norm estimates. This will yield Proposition 4.11, which will be used in Section 5 to prove Proposition 3.1.
4.1 Abstract Wiener spaces
If H is a real separable infinite-dimensional Hilbert space, it is impossible to define on its Borel σ-algebra a measureµH,h such that (7) holds with B instead ofH.
Nevertheless, one can define a promeasure (or cylindrical probability measure in the sense of [K-R]) µH,h
on the cylinder sets ofH. For everyE in F(H) and every positiveh, one can define a Gaussian measureµE,h
on E by (5), where λE is the Lebesgue measure on E (normalized in a natural way) and | · | is the norm of H. A cylinder set of H is any set of the form C=π−1E (Ω), where E ∈ F(H),πE :H →E is the orthogonal projection and Ω is a Borel set ofE. If C is such a cylinder set, one setsµH,h(C) =µE,h(Ω). In other words, for every Borel set Ω ofE:
µH,h(π−1E (Ω)) = (2πh)−dim(E)/2 Z
Ω
e−|y|
2
2h dλ(y). (42)
One defines this way an additive set function on the cylinder sets ofH, but if H is infinite-dimensional, this function is not σ-additive andµH,h does not extend as a measure on the σ-algebra generated by the cylinder sets (which is the Borelσ-algebra).
If the Hilbert space H is included into a Banach space B, the canonical injection being continuous and having a dense range, so that (4) is satisfied, one can define the cylinder sets of B as the sets C = PE−1(Ω), where E∈ F(B0), wherePE :B →E is defined by (6) and where Ω is a Borel set ofE. One then defines an additive set functionµB,h on the cylinder sets ofB as in (42):
µB,h(PE−1(Ω)) = (2πh)−dim(E)/2 Z
Ω
e−|y|
2 2h dλ(y).
But this time, ifB is well chosen, the additive sets function µB,hextends as a measure on the Borelσ-algebra ofB. The following definition specifies the conditions whichB must fulfill.
Definition 4.1. [G-2, G-3, K] An abstract Wiener space is a triple (i, H, B) where H is a real separable Hilbert space,B a Banach space andia continuous injection fromH intoB, such thati(H)is dense inB and satisfying, moreover, the following condition. For all positiveεand h, there exists a subspace F inF(H)such that, for allE inF(H), orthogonal toF,
µH,h({x∈H, ki(πE(x))kB > ε})< ε.
The norm on B is said to be “measurable”.
If (i, H, B) is an abstract Wiener space (in other words if the norm ofB is measurable in the sense above) and if (4) holds one proves that, for every positiveh, the additive set functionµB,h, defined on the cylinder set functions of B, extends as a measure on the Borel σ-algebra of B and has the following property. For every finite system {u1, ..., un} of B0, which is orthonormal with respect to the scalar product of H, the functions x→uj(x) (defined onB) are independent Gaussian random variables and the system{u1, ...un}has the normal distributionµRn,h. See [G-2], [G-3] and [K] (consequence of the Theorems 4.1 and 4.2, Chapter 1). For every E inF(B0) and everyϕinL1(E, µE,h), the equality (7) is satisfied, according to the transfer Theorem.
Let us recall a classical example of the explicit computation of an integral, where a is in the complexified spaceHC:
Z
B
e`a(x)dµB,h(x) =eha
2
2 . (43)
One has seta2=|u|2− |v|2+ 2iu·v ifa=u+iv, withuandv inH. Let us recall, too, that for allainHand for allp≥1:
Z
B
|`a(x)|pdµB,h(x) =(2h)p/2
√π |a|p Γ p+ 1
2
. (44)
One sees, too, that for allaandB in H, Z
B
e`b(u)|`a(u)|pdµB,h(u) =eh|b|
2 2
Z
R
|√
h|a|v+ha·b|pdµR,1(v). (45) The following proposition allows to deal with translations by a vectorabelonging toH. There is no such result for a translation by a vectorabelonging toB.
Proposition 4.2. [K], p 113,114 Let (i, H, B) be an abstract Wiener space and µB,h its measure. For all g∈L1(B, µB,h), one has, for all ainH:
Z
B
g(x)dµB,h(x) =e−2h1|a|2 Z
B
g(x+a)e−h1`a(x)dµB,h(x). (46) For everya∈H and everyf in L2(B, µB,h), one has
Z
B
|f(x)|2dµB,h(x) =e−2h1|a|2 Z
B
|f(x+a)|2e−h1`a(x)dµB,h(x). (47) Let us recall the theorem of Wick :
Theorem 4.3. WickLet u1, ...u2p be vectors ofH (p≥1). Let h >0. Then one has Z
B
`u1(x)...`u2p(x)dµB,h(x) =hp X
(ϕ,ψ)∈Sp
p
Y
j=1
< uϕ(j), uψ(j)> (48)
whereSp is the set of all couples(ϕ, ψ)of injections from{1, ..., p}into{1, ....,2p} such that:
1. For allj≤p,ϕ(j)< ψ(j).
2. The sequence(ϕ(j))(1≤j≤k) is an increasing sequence.
One deduces from (43) the following inequalities, which hold for all aandbin the complexified ofH : Z
B
e`a(x)−e`b(x)
2
dµB,h(x)≤4h|a−b|(|a|+|b|)e2hmax(|Rea|2,|Reb|2) Z
B
|e`a(x)−e`b(x)|2 dµB,h(x)≤e2hmax(|Re(a)|,|Re(b)|)2h|a−b|2(1 + 4hmax(|Re(a)|,|Re(b)|)2)
(49)
4.2 Stochastic extensions
In order to define the stochastic extension of a function f : H → C, we first define the extension of the orthogonal projectionπE :H →E, whereE∈ F(H).
With each E inF(H), one can associate a mapeπE :B→E defined almost everywhere by
πeE(x) =
dim(E)
X
j=1
`uj(x)uj, (50)
where the uj (1 ≤j ≤dim(E)) form an orthonormal basis of E and `uj is defined in Proposition 1.1. This map does not depend on the choice of the orthonormal basis and, whenE⊂B0, it coincides with the mapPE
already defined by (6).
Remark that (7) still holds for the subspaces E in F(H) (and not necessarily in F(B0)), provided PE is replaced byeπE. See Lemma 4.7 in [K].
Notice that, for every subspace E inF(H):
a·(πeE(x)) =`πE(a)(x) a∈H a.e.x∈B (51)
The following notion has been introduced by Gross [G-1], who gives conditions for the existence of the extension. Other conditions or examples will be found in Section 8 or in [A-J-N-1].
Definition 4.4. [G-1, G-2, G-3] [RA] [K] Let (i, H, B) be an abstract Wiener space satisfying (4). (The inclusion iwill be omitted). Lethbe a positive real number.
1. A Borel function f, defined onH, is said to admit a stochastic extension fewith respect to the measure µB,h if, for every increasing sequence(En)inF(H), whose union is dense inH, the sequence of functions f ◦πeEn (where eπEn is defined by (50)) converges in probability with respect to the measure µB,h tofe. In other words, if, for everyδ >0,
n→+∞lim µB,h
n
x∈B, |f◦eπEn(x)−fe(x)|> δo
= 0. (52)
2. A functionf is said to admit a stochastic extensionfe∈Lp(B, µB,h)in the sense ofLp(B, µB,h)(1≤p <
∞) if, for every increasing sequence(En)inF(H), whose union is dense inH, the functionsf◦πeEn are in Lp(B, µB,h)and if the sequencef ◦πeEn converges inLp(B, µB,h)tofe.
One defines likewise the stochastic extension of a function onH2 to a function onB2.
Iffeis the stochastic extension of a functionf :H →H, one cannot say thatf is the restriction offetoH. SinceH is negligible (see Kuo [K]), this is irrelevant. For everyain H one sees that the applicationu→u·a, defined on H, admits a stochastic extension which is the function `a. This is a consequence of the equality (51). In Definition 4.4, the functions can take their values in a Banach space. Hence one can say that the application eπE of (50) is the stochastic extension of the orthogonal projection πE :H →E. One will find in Section 8 examples of functions admitting stochastic extensions. In particular, if a function f is bounded on H and uniformly continuous with respect to the restriction toH of the norm ofB, then it admits a stochastic extensionfe, which coincides with its density extension (see Kuo [K], Chapter 1, Theorem 6.3).
If a Borel functionf is bounded onH, does not depend onhand admits, for every positiveh, a stochastic extension with respect toµB,h, this extension may depend onh. It is not the case in the situation of Proposition 8.2. In the other cases, we may consider that the stochastic extension is independent ofhifhvaries in a countable subsetQof (0,+∞).
4.3 The Heat operator (continued)
In the rest of this work, (i, H, B) represents an abstract Wiener space satisfying (4) and the injectioniwill be omitted. We complete the investigation begun in Sections 2.1 and 2.3.
LetE be inF(B0), letE⊥ be its orthogonal space, defined in (24) and lett >0. For every Borel function F bounded onB2, letHeE⊥,tF be the function defined in (27):
(HeE⊥,tF)(X) = Z
(E⊥)2
F(X+Y)dµ(E⊥)2,t(Y).
